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<h1>bioheatExact</h1>
<p class="purpose">Compute exact solution to Pennes' bioheat equation in homogeneous media.</p>

<h2>Syntax</h2>

<pre class="codeinput">
T = bioheatExact(T0, S, material, dx, t)
T = bioheatExact(T0, S, [D, P, Ta], dx, t)
</pre>

<h2>Description</h2>
<p><code>bioheatExact</code> calculates the exact solution to Pennes' bioheat equation in a homogeneous medium on a uniform Cartesian grid using a Fourier-based Green's function solution assuming a periodic boundary condition [1]. The function supports inputs in 1D, 2D, and 3D. The exact equation solved is given by</p>

<pre class="codeinput">
dT/dt = D * d^2T/dx^2 - P * (T - Ta) + S
</pre>

<p>where the coefficients are defined below. Pennes' bioheat equation is often given in the alternative form</p>

<pre class="codeinput">
P0 * C0 * dT/dt =  Kt * d^2T/dx^2 - Pb * Wb * Cb * (T - Ta) + Q

T:  temperature                     [degC]
C0: tissue specific heat capacity   [J/(kg.K)]
P0: tissue density                  [kg/m^3] 
Kt: tissue thermal conductivity     [W/(m.K)]
Pb: blood density                   [kg/m^3] 
Wb: blood perfusion rate            [1/s]
Ta: blood arterial temperature      [degC]
Cb: blood specific heat capacity    [J/(kg.K)]
Q:  volume rate of heat deposition  [W/m^3]
</pre>

<p>In this case, the function inputs are calculated by</p>

<pre class="codeinput">
D = Kt / (P0 * C0);
P = Pb * Wb * Cb / (P0 * C0);
S = Q / (P0 * C0);
</pre>

<p>If the perfusion coefficient <code>P</code> is set to zero, <code>bioheatExact</code> calculates the exact solution to the heat equation in a homogeneous medium.<p>

<p>[1] Gao, B., Langer, S., & Corry, P. M. (1995). Application of the time-dependent Green's function and Fourier transforms to the solution of the bioheat equation. International Journal of Hyperthermia, 11(2), 267-285.</p>

<h2>Inputs</h2>

<table class="body">
    <tr valign="top">
        <td width = "150"><code>T0</code></td>
        <td>matrix of the initial temperature distribution at each grid point [degC]</td>
    </tr>
    <tr valign="top">
        <td><code>S</code></td>
        <td>matrix of the heat source at each grid point [K/s]</td>        
    </tr>
	<tr valign="top">
        <td><code>material</code></td>
        <td>material coefficients given as a three element vector in the form: <code>material = [D, P, Ta]</code>, where
        
		<table>
			<tr valign="top">
				<td><code>D</code></td>
				<td>diffusion coefficient [m^2/s]</td>
			</tr>
			<tr valign="top">
				<td><code>P</code></td>
				<td>perfusion coefficient [1/s]</td>        
			</tr>
			<tr valign="top">
				<td><code>Ta</code></td>
				<td>arterial temperature [degC]</td>        
			</tr>
		</table>
        
        </td>        
    </tr>
	<tr valign="top">
        <td><code>dx</code></td>
        <td>grid point spacing [m]</td>        
    </tr>
	<tr valign="top">
        <td><code>t</code></td>
        <td>time at which to calculate the temperature field [s]</td>
    </tr>
</table>

<h2>Outputs</h2>

<table class="body">
    <tr valign="top">
        <td width = "150"><code>T</code></td>
        <td>temperature field at time <code>t</code> [degC]</td>
    </tr>
</table>

<h2>See Also</h2>

<code><a href="kWaveDiffusion.html">kWaveDiffusion</a></code>

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